Population genetics
Population genetics studies how inherited variants are distributed in populations and how their frequencies change through evolutionary processes.
Alleles and genotypes
Consider one diploid locus with two alleles, \(A\) and \(a\). The possible genotypes are
\[AA,\qquad Aa,\qquad aa.\]Let their population frequencies be
\[f_{AA},\qquad f_{Aa},\qquad f_{aa}.\]Because every individual belongs to one of these three genotype classes,
\[\boxed{f_{AA}+f_{Aa}+f_{aa}=1}.\]Allele frequencies from genotype frequencies
Let \(p\) be the frequency of allele \(A\) and \(q\) the frequency of allele \(a\). Each \(AA\) individual contributes only \(A\) alleles, while each heterozygote contributes one \(A\) and one \(a\). Therefore
\[\boxed{p=f_{AA}+\frac12f_{Aa}},\]\[\boxed{q=f_{aa}+\frac12f_{Aa}}.\]Hence
\[\boxed{p+q=1}.\]Counting alleles directly
Suppose a diploid population contains \(N\) individuals, with counts \(N_{AA}\), \(N_{Aa}\) and \(N_{aa}\). There are \(2N\) allele copies in total. The number of \(A\) copies is
\[2N_{AA}+N_{Aa}.\]Thus
\[\boxed{p=\frac{2N_{AA}+N_{Aa}}{2N}}.\]Similarly,
\[\boxed{q=\frac{2N_{aa}+N_{Aa}}{2N}}.\]A simple example
If
\[f_{AA}=0.36,\qquad f_{Aa}=0.48,\qquad f_{aa}=0.16,\]then
\[p=0.36+\frac12(0.48)=0.60,\]and
\[q=0.16+\frac12(0.48)=0.40.\]The genotype distribution contains more information than the allele frequencies alone: several different genotype distributions can have the same value of \(p\).
What can change allele frequencies?
| Process | Population-genetic effect |
|---|---|
| Selection | Types with different reproductive success contribute unequally to future generations. |
| Mutation | Alleles are converted into other allelic states. |
| Migration or gene flow | Individuals or gametes move genetic variants between populations. |
| Genetic drift | Finite-population sampling changes frequencies randomly. |
| Recombination | Associations between alleles at different loci are rearranged. |
These processes do not all act in the same mathematical way. Later lessons isolate several of them so their effects can be understood separately before they are combined.
Allele frequency versus population size
An allele can increase in frequency even while the total population declines. Conversely, the number of copies of an allele can increase while its frequency decreases if other types increase faster.
One generation as a mathematical update
A discrete-generation population-genetic model often has the form
\[\boxed{p_{n+1}=F(p_n)},\]where the function \(F\) represents the evolutionary mechanisms acting during one generation.
For example, the previous lesson showed that simple haploid selection gives
\[p_{n+1}=\frac{p_nw_A}{p_nw_A+(1-p_n)w_a}.\]Other mechanisms produce different update rules.
Deterministic models
A deterministic model assigns a definite next frequency once the current state and parameters are known. Such models are useful when population size is large enough that random sampling is relatively small or when the aim is to understand the systematic effect of selection, mutation or migration.
Stochastic models
In a finite population, the next generation is formed from a finite number of reproductive outcomes. Even if every individual follows the same probabilistic rules, the realised allele frequency can vary between replicate populations.
This random change is genetic drift. The later Wright–Fisher and Moran lessons develop explicit stochastic models for it.
Fixation and loss
For a two-allele system,
\[p=1\]means allele \(A\) is fixed, while
\[p=0\]means it has been lost. In models without mutation or migration, these boundary states are often absorbing for the allele.
Multiple alleles
If a locus has alleles \(A_1,\ldots,A_m\) with frequencies \(p_1,\ldots,p_m\), then
\[\boxed{\sum_{i=1}^{m}p_i=1}.\]The two-allele relation \(q=1-p\) is therefore a special case of a more general frequency constraint.
More than one locus
When several loci are modelled, allele frequencies alone may not describe the population completely. Associations between alleles at different loci can matter, and recombination can change those associations even when individual allele frequencies remain unchanged.
This leads to concepts such as haplotype frequency and linkage disequilibrium in more advanced population genetics.
From allele frequencies to genotype frequencies
Knowing \(p\) and \(q\) does not by itself determine the genotype frequencies unless assumptions are added about how alleles are paired into individuals.
Under random mating and the classical Hardy–Weinberg assumptions, the genotype frequencies become
\[p^2,\qquad 2pq,\qquad q^2.\]The next lesson derives this result and explains exactly what the equilibrium does—and does not—mean.